3 Answers
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I suppose the best way to do this is to show the two inclusions (namely $f(X\cup Y)\subseteq f(X)\cup f(Y)$ and $f(X)\cup f(Y)\subseteq f(X\cup Y)$), and to use definitions. For the first inclusion, this gives :

Let $x\in f(X\cup Y)$. This means that there exists a $y\in X\cup Y$ such that $f(y)=x$. Now $y\in X\cup Y$ means that either $y\in X$ or $y\in Y$. Now if $y\in X$, then $x\in f(X)$, and if $y\in Y$, then $x\in f(Y)$. In any instance, $x\in f(X)\cup f(Y)$. We proved that any $x$ in $f(X\cup Y)$ is also in $f(X)\cup f(Y)$, that is precisely $f(X\cup Y)\subseteq f(X)\cup f(Y)$. The other inclusions proceeds similarly.

$\LaTeX$ tip: use \subset or \subseteq for set inclusion, not \in. We want to show $f(X\cup Y)\subseteq f(X)\cup f(Y)$, not $f(X\cup Y)\in f(X)\cup f(Y)$ (which could be true, but is unlikely)
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Arturo MagidinMar 23 '12 at 15:17

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@Dimitri: It would have been good for the OP to make the correction himself, given that he's new and this is his first post.
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Arturo MagidinMar 23 '12 at 15:33

To get started, write out the definitions of $f(X\cup Y)$, $f(X)$, and $f(Y)$:
$$f(X\cup Y)=\{b\in B\mid \text{there exists an }a\in X\cup Y\text{ such that }b=f(a)\}$$
$$f(X)=\{b\in B\mid \text{there exists an }a\in X\text{ such that }b=f(a)\}$$
$$f(Y)=\{b\in B\mid \text{there exists an }a\in Y\text{ such that }b=f(a)\}$$
Also, remember that for two sets $C$ and $D$,
$$z\in C\cup D\iff z\in C\;\text{ or }\,z\in D.$$
Do you see how to go the rest of the way?

The simplest method two prove two sets are equal is to show that each one is contained in the other.

The simplest method to show that one set is contained in the other is to show that any element in the one set is also an element in the other.

Here you will want to show that
$$f(X\cup Y)\subseteq f(X)\cup f(Y)\quad\text{and}\quad f(X)\cup f(Y)\subseteq f(X\cup Y)$$
both hold.

For example, to show that $f(X\cup Y)\subseteq f(X)\cup f(Y)$, let $b\in f(X\cup Y)$. We need to show that $b\in f(X)\cup f(Y)$; that is, we need to show that either $b\in f(X)$, or $b\in f(Y)$.

Since $b\in f(X\cup Y) = \{f(a)\mid a\in X\cup Y\}$, there exists $a\in X\cup Y$ such that $b=f(a)$. Since $a\in X\cup Y$, either $a\in X$ or $a\in Y$. If $a\in X$, then $b=f(a)\in f(X)\subseteq f(X)\cup f(Y)$, and we are done. If $a\in Y$, then $b=f(a)\in f(Y)\subseteq f(X)\cup f(Y)$, and we are done. Since these are the only cases, we see that if $b\in f(X\cup Y)$, then $b\in f(X)\cup f(Y)$. This proves the first inclusion.